Why does the rendered and baked lighting differ so much?
All I did was switch to CPU (I have GPU problems on my Mac) and decrease render samples to speed up the bake and I got this:



My thought is that perhaps your problem was with the GPU since that is the only relevant thing that I changed (depending on what GPU you have, some features of Blender don't work, for example, HDR lighting). 

Here is the baked image with higher samples: